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I'm trying to integrate the following expression:

f[a1,p1+p2+p3+p4]*f[a2,p2+p3+p4]*f[a3,p3+p4]*f[a4,p4]

Where

f[x_, u_] := E^(-(x - u)/2)/(2*(1 + E^(-(x - u)/2))^2)

i.e. the PDF of the logistic distribution with s=2.

Also, the a's are positive integers.

The tricky part is that I want to integrate over all values of the p values that result in them forming a valid probability distribution, ie the p variables must be greater than zero, less than one and sum to 1.

I've tried using a definite integral with the following limits (after substituting p4 with $1-p1-p2-p3$:

{p1,0,1},{p2,0,1-p1},{p3,0,1-p2-p3}

This works for a simplified version of the problem (with 3 or 2 "f" terms and p variables but as soon as I use 4 it takes about 20 minutes and then just returns the input. My instinct is that if it can be analytically found for 3 terms, 4 terms should also be findable, just with more terms.

My question(s) are:

1) Does anyone have any suggestions to get Mathematica working for this (and larger problems)?

2) I'm not tied to the logistic distribution, I'm using it because I thought it's simple CDF would make the problem easy for Mathematica. Can anyone suggest a similar shaped distribution that would make this problem tractable?

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I think these limits don't ensure $0 \leq p_i \leq 1$ and $\sum_{i} p_i =1$. –  Mahdi Jun 5 at 4:02
    
Are you sure? That's what I saw other people solving similar problems (eg this post –  Amit Deshwar Jul 5 at 16:29

1 Answer 1

I think the problem is in limits you use because they don't ensure $0 \leq p_i \leq 1$ and $\sum_{i} p_i =1$. You have 4 $p_i$'s so you can only fix $p_4$. By using this relation, we can write the integrand as:

eq= f[a2, 1 - p1]*f[a3, 1 - p1 - p2]*f[a4, 1 - p1 - p2 - p3]*f[a1,1];

The last terms does not depend on $p_i$'s. Now you can easily find indefinite solution:

g=Integrate[eq,p1,p2,p3];

which is a function such that $g=g(p_1,p_2,p_3)$ with only one constraint $0 \leq p_1+p_2+p_3 \leq 1$. Then basically you can have value of $g$ over any valid combinations of $p_i$'s.

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So once I've done this and I have g, then I still need to integrate over p1,p2,p3 with the constraint that they're positive and are less than one? –  Amit Deshwar Jul 5 at 16:33

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